| L(s) = 1 | − 37.3·2-s − 245.·3-s + 884.·4-s − 1.54e3·5-s + 9.18e3·6-s − 9.37e3·7-s − 1.39e4·8-s + 4.07e4·9-s + 5.79e4·10-s − 3.18e4·11-s − 2.17e5·12-s − 3.83e3·13-s + 3.50e5·14-s + 3.80e5·15-s + 6.75e4·16-s − 1.52e6·18-s + 7.04e5·19-s − 1.37e6·20-s + 2.30e6·21-s + 1.19e6·22-s − 1.41e5·23-s + 3.42e6·24-s + 4.47e5·25-s + 1.43e5·26-s − 5.16e6·27-s − 8.29e6·28-s + 5.40e6·29-s + ⋯ |
| L(s) = 1 | − 1.65·2-s − 1.75·3-s + 1.72·4-s − 1.10·5-s + 2.89·6-s − 1.47·7-s − 1.20·8-s + 2.06·9-s + 1.83·10-s − 0.655·11-s − 3.02·12-s − 0.0372·13-s + 2.43·14-s + 1.94·15-s + 0.257·16-s − 3.41·18-s + 1.24·19-s − 1.91·20-s + 2.58·21-s + 1.08·22-s − 0.105·23-s + 2.10·24-s + 0.228·25-s + 0.0615·26-s − 1.87·27-s − 2.55·28-s + 1.41·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.3019754114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3019754114\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 37.3T + 512T^{2} \) |
| 3 | \( 1 + 245.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.54e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 9.37e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.18e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.83e3T + 1.06e10T^{2} \) |
| 19 | \( 1 - 7.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.41e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.40e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.78e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.94e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.41e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.87e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.19e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.33e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.05e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 9.79e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.49e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 7.01e6T + 4.58e16T^{2} \) |
| 73 | \( 1 + 9.11e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 8.25e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.73e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.66e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.74e8T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.18024027093566446868163499932, −9.638239752143314428964034105701, −8.269332544748013010180586812200, −7.27892934675817693227127772390, −6.68453422245624334868227114593, −5.67729045167157546984306396631, −4.29424022114903160804906009698, −2.79538526909156345803401728809, −0.797341094327779799519936849970, −0.55580749060222986739554207473,
0.55580749060222986739554207473, 0.797341094327779799519936849970, 2.79538526909156345803401728809, 4.29424022114903160804906009698, 5.67729045167157546984306396631, 6.68453422245624334868227114593, 7.27892934675817693227127772390, 8.269332544748013010180586812200, 9.638239752143314428964034105701, 10.18024027093566446868163499932
